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{{MSC|60B10}}

Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures)  $  \{ P _ {n} \} $
on the Borel sets of a metric space  $  S $
is called weakly convergent to a distribution  $  P $
if

$$ \tag{* }
\lim\limits _ { n }  \int\limits _ { S } f  dP _ {n}  =  \int\limits _ { S } f  dP
$$

for any real-valued bounded continuous function  $  f $
on  $  S $. 
Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign  $  \Rightarrow $. 
The following conditions are equivalent to weak convergence:

1) (*) holds for any bounded uniformly-continuous real-valued function  $  f $;

2) (*) holds for any bounded  $  P $-
almost-everywhere continuous real-valued function  $  f $;

3)  $  \lim\limits _ {n} \sup  P _ {n} ( F) \leq  P ( F) $
for any closed set  $  F \subset  S $;

4)  $  \lim\limits _ {n} \inf  P _ {n} ( G) \geq  P ( G) $
for any open set  $  G \subset  S $;

5)  $  \lim\limits _ {n}  P _ {n} ( A) = P ( A) $
for any Borel set  $  A \subset  S $
with  $  P ( \partial  A) = 0 $, 
where  $  \partial  A $
is the boundary of  $  A $;

6)  $  \lim\limits _ {n}  p ( P _ {n} , P) = 0 $, 
where  $  p $
is the [[Lévy–Prokhorov metric|Lévy–Prokhorov metric]].

Let  $  U $
be a class of subsets of  $  S $, 
closed under intersection and such that every open set in  $  S $
is a finite or countable union of sets in  $  U $. 
Then if  $  \lim\limits _ {n}  P _ {n} ( A) = P ( A) $
for all  $  A \in U $, 
it follows that  $  P _ {n} \Rightarrow P $. 
If  $  S = \mathbf R  ^ {k} $
and  $  F _ {n} $, 
$  F $
are the distribution functions corresponding to  $  P _ {n} $, 
$  P $
respectively, then  $  P _ {n} \Rightarrow P $
if and only if  $  F _ {n} ( x) \rightarrow F ( x) $
at every point  $  x $
where  $  F $
is continuous.

Let  $  S $
be a [[Separable space|separable space]] and let  $  {\mathcal F} $
be the class of real-valued bounded Borel functions on  $  S $. 
To have  $  \int _ {S} f  dP _ {n} \rightarrow \int _ {S} f  dP $
uniformly over  $  f \in {\mathcal F} $
for every sequence  $  \{ P _ {n} \} $
such that  $  P _ {n} \Rightarrow P $, 
it is necessary and sufficient that:

a)

$$ 
\sup _ {f \in F }  \omega _ {f} ( S)  <  \infty ,
$$

b)

$$ 
\lim\limits _ {\epsilon \downarrow 0 }  \sup _ {f \in {\mathcal F} } \ 
P ( \{ {x } : {\omega _ {f} ( S _ {x, \epsilon }  ) > \delta } \}
)  =  0 \ \ 
\textrm{ for }  \textrm{ all }  \delta > 0,
$$

where

$$ 
\omega _ {f} ( A)  = \ 
\sup \ 
\{ {| f ( x) - f ( y) | } : {x, y \in A } \}
$$

and  $  S _ {x, \epsilon }  $
is the open ball of radius  $  \epsilon $
with centre  $  x $. 
If the class  $  {\mathcal F} $
is generated by the indicator functions of sets from some class  $  E $, 
then conditions a) and b) lead to the condition

$$ 
\lim\limits _ {\epsilon \downarrow 0 } \ 
\sup _ {A \in E } \ 
P ( A  ^  \epsilon  \setminus 
A ^ {- \epsilon } )  =  0,
$$

where

$$ 
A  ^  \epsilon   = \ 
\cup _ {x \in A }
S _ {x, \epsilon }  ,\ \ 
A ^ {- \epsilon }  = \ 
S \setminus  ( S \setminus  A)  ^  \epsilon 
$$

(when each open ball in  $  S $
is connected,  $  A  ^  \epsilon  \setminus  A ^ {- \epsilon } = ( \partial  A)  ^  \epsilon  $). 
If  $  S = \mathbf R  ^ {k} $
and the distribution  $  P $
is absolutely continuous with respect to Lebesgue measure, then  $  P _ {n} \Rightarrow P $
if and only if  $  P _ {n} ( A) \rightarrow P ( A) $
uniformly over all convex Borel sets  $  A $.

Let  $  P _ {n} $, 
$  P $
be distributions on a metric space  $  S $
such that  $  P _ {n} \Rightarrow P $
and let  $  h $
be a continuous  $  P $-
almost-everywhere measurable mapping of  $  S $
into a metric space  $  S  ^  \prime  $. 
Then  $  P _ {n} h  ^ {-} 1 \Rightarrow Ph  ^ {-} 1 $, 
where for any distribution  $  Q $
on  $  S $, 
the distribution  $  Qh  ^ {-} 1 $
is its  $  h $-
image on  $  S  ^  \prime  $:

$$ 
Qh  ^ {-} 1 ( A)  =  Q ( h  ^ {-} 1 ( A))
$$

for any Borel set  $  A \in S  ^  \prime  $.

A family  $  {\mathcal P} $
of distributions on  $  S $
is said to be weakly relatively compact if every sequence of elements of it contains a weakly convergent subsequence. A condition for weak relative compactness is given by Prokhorov's theorem. A family  $  {\mathcal P} $
is called tight if, for any  $  \epsilon > 0 $, 
there is a compact set  $  K \subset  S $
such that  $  P ( K) > 1 - \epsilon $, 
for all  $  P \in {\mathcal P} $. 
Prokhorov's theorem now states: If  $  {\mathcal P} $
is tight, then it is relatively compact; if, moreover,  $  S $
is separable and complete, then weak relative compactness of  $  {\mathcal P} $
implies its tightness. In the case when  $  S = \mathbf R  ^ {k} $, 
a family  $  {\mathcal P} $
of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to  $  {\mathcal P} $
is equicontinuous at zero.

Now let  $  P _ {n} $, 
$  P $
be distributions on a measure space  $  ( X, A) $, 
where  $  A $
is a  $  \sigma $-
algebra. Convergence in variation of  $  P _ {n} $
to  $  P $
means uniform convergence on all sets in  $  A $
or, equivalently, convergence on all sets in  $  A $
or, equivalently, convergence of the variation

$$ 
| P _ {n} - P |  = \ 
( P _ {n} - P)  ^ {+} +
( P _ {n} - P)  ^ {-}
$$

to zero. Here,  $  ( P _ {n} - P)  ^ {+} $
and  $  ( P _ {n} - P)  ^ {-} $
are the components in the Jordan–Hahn decomposition of the signed measure  $  P _ {n} - P $.

====References====
{|
|valign="top"|{{Ref|B}}|| P. Billingsley, "Convergence of probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}}
|-
|valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , Princeton Univ. Press (1963) {{MR|0203748}} {{ZBL|0108.14202}}
|-
|valign="top"|{{Ref|BR}}|| R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions" , Wiley (1976) {{MR|0436272}} {{ZBL|}}
|}

====Comments====
For more information on weak convergence see [[Weak convergence of probability measures|Weak convergence of probability measures]]; [[Convergence of measures|Convergence of measures]].